Constraint Dependence of Active Depletion Forces on Passive Particles

Peng Liu,1,2,* Simin Ye,1,2,* Fangfu Ye,1,2,3,4,† Ke Chen,1,2,4,‡ and Mingcheng Yang1,2,§1Beijing National Laboratory for Condensed Matter Physics and Laboratory of Soft Matter Physics, Institute of Physics,

Chinese Academy of Sciences, Beijing 100190, China2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

3Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China4Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China

(Received 6 September 2019; revised manuscript received 22 February 2020; accepted 25 March 2020; published 13 April 2020)

Using simulations and experiments, we demonstrate that the effective interaction between passiveparticles in an active bath substantially depends on an external constraint suffered by the passive particles.Particularly, the effective interaction between two free passive particles, which is directly measured insimulation, is qualitatively different from the one between two fixed particles. Moreover, we find that thefriction experienced by the passive particles—a kinematic constraint—similarly influences the effectiveinteraction. These remarkable features are in significant contrast to the equilibrium cases, and mainly arisefrom the accumulation of the active particles near the concave gap formed by the passive spheres. Thisconstraint dependence not only deepens our understanding of the “active depletion force,” but also providesan additional tool to tune the effective interactions in an active bath.

DOI: 10.1103/PhysRevLett.124.158001

Introduction.—Active matters consisting of self-propelled units are an important class of nonequilibriumsystems, ranging from cell cytoskeleton [1,2] to biologicaland artificial microswimmers [3–12] to shaken grains [13–16]. Active matter systems often exhibit exotic behaviorsnot found in passive systems, including complex collectivemotions [17–24], motility-induced phase separation [25–27], or wall-dependent pressure [28,29]. Apart from theemergence of intriguing nonequilibrium phenomena, anactive bath can significantly affect immersed passiveobjects compared to an equilibrium bath. For instance,active environment can enhance diffusion and effectivetemperature of tracer particles [30–32], power a directedmotion of asymmetric objects [33–36], dramaticallydeform flexible macromolecules [37–42], and induceunexpected self-assemblies and phase separations of pas-sive colloids [43–47]. To exploit these nonequilibriumbehaviors of immersed passive particles, it is fundamentallyimportant to understand interparticle effective interactionmediated by the active bath, i.e., active depletion force.The active depletion force is a conceptual generalization

of the equilibrium depletion force [48–53] that is aneffective interaction between large particles suspended inan overwhelming number of smaller particles or depletants,and arises from the gain of the system entropy dominatedby the depletants by compressing the phase space of thelarge particles. In active systems, however, this entropyscenario is no longer applicable.Recently, the active depletion interactions are either

characterized from pair correlation function, gðrÞ of uncon-strained passive particles [44,54–56] or determined by

measuring the forces, FeffðrÞ, exerted on two fixed objectsby the active bath [29,55,57–63]. In equilibrium systems,the effective force and the pair correlation function aredirectly related by FeffðrÞ ¼ kBT∇ ln½gðrÞ� − FrðrÞ, withFrðrÞ the steric interaction between passive particles. Thisrelation, however, is likely to fail in an active bath, which isfar from equilibrium. More importantly, it is not clear at allwhether either method obtains the true effective forcesexperienced by free passive particles in an active bath, thedriving force for self-assemblies or phase behaviors in thenonequilibrium systems.In this Letter, we directly measure the active depletion

forces between two passive particles in steady states, usingcomputer simulations and laser tweezer experiments. Wefind that the effective force on the free passive particleclearly deviates from that inferred from gðrÞ. By con-straining the passive particles under external traps, wedemonstrate that the active depletion force sensitivelydepends on the degree of the constraint, thus the FeffðrÞon a free object is qualitatively different from the fixedcase. In addition, the active depletion forces for free passiveobjects vary with their frictional coefficients that imposekinematic constraints. Microscopically, the constraint-dependent effective forces come from the fact that thedistribution of active depletant near passive objects dependssensitively on the constraints of the objects. Our results thusindicate that the active depletion is much more complexthan expected and cannot be treated in the same fashion asin equilibrium systems.Simulation.—We consider a 2D system consisting of

1000 small self-propelled particles of diameter σs and two

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large passive particles of diameter σl ¼ 3σs, as sketched inFig. 1(a). All particles interact with each other through arepulsive Lennard-Jones type of potential, UðrÞ ¼4ϵ½ðσ=rÞ24 − ðσ=rÞ12� þ ϵ for r < 21=12σ. Here, the inter-action diameter between the passive and active particles istaken as σ ¼ ðσs þ σlÞ=2. The translational dynamics ofthe active particles is described by overdamped Langevinequation, γsv ¼ Fd þ Fr þ η, where γs is the translationalfriction coefficients, Fd the self-propelling force, Fr thesteric interaction between particles, and η the Gaussian-distributed stochastic force of zero mean and variancehηðtÞηðt0Þi ¼ 2kBTγsδðt − t0Þ with kBT ¼ ϵ. The orienta-tion of the active particles evolves according to rotationaldiffusion, with the rotational frictional coefficient γr ¼13σ2sγs unless otherwise stated. The dynamics of the

isotropic passive particles also obeys an overdampedLangevin equation, with the frictional coefficient γland Fd ¼ 0. When considering the passive particles inconfinement, an external trapping potential, UexðrÞ ¼12kelðr − r0Þ2, is applied with r0 the trap center.Measurement of the depletion force.—In simulations, the

FeffðrÞ is directly measured by accumulating the averageforce on the free passive particles exerted by the activeparticles in a small range of separation around r in thesteady state. Generally, this accumulated mean forceincludes the total depletion force, and partial frictionaland stochastic forces experienced by the passive particle(thermal bath also contributes to frictional and stochasticforces), which prevents the extraction of FeffðrÞ. However,when particle distribution reaches the steady state, thepassive particle flux and hence the friction vanish. Theaveraged stochastic force is also zero at steady state. In thenonequilibrium steady state, the nonuniform pressure andswimmer density, arising from the anisotropic interactionsbetween the active swimmers and the passive particle pair,may spontaneously generate a local current of swimmers[40]. This current, however, does not contribute to the

friction on the passive particles, as it is self-generatedinstead of externally imposed and is an accompanyingeffect of the active depletion interaction. Thus, the averageforce measured at steady states can be exclusively attrib-uted to the active depletion force. The method is valid inboth equilibrium and nonequilibrium systems, and has beendemonstrated in simulations by computing equilibriumdepletion forces (see the Supplemental Material [64])and thermophoretic force in nonisothermal solutions[71]. This procedure can also be used to measure FeffðrÞon passive particles in elastic traps when the passiveparticles fluctuate close to their equilibrium positions.Fixing passive objects, corresponding to infinitely strongtraps, also eliminates the friction but it drastically changesthe particle dynamics, which may affect the active depletioninteractions.Experiment.—A dilute dispersion of polystyrene (PS)

beads of diameter σl ¼ 3 μm is mixed with a solution ofE.coli bacteria (σs ¼ 0.6 μm in diameter, 4σs in length,with a swimming speed ∼20 μm=s), and is then loaded intoa glass cell with a thickness of 30 μm. Two PS particles aretrapped near the upper cover slide by two identical opticaltweezers (Aresis Tweez 250si) at the same height, assketched in Fig. 1(b). The laser intensity is low such thatthe number density and motility of the bacteria are hardlyaffected by the irradiation. The PS particles are confined tofluctuate around a separation rwithout direct contact. In thesteady state, the active depletion forces on the particlesfrom the surrounding bacteria shift the particle equilibriumpositions slightly away from the trap centers. Each particlemay also experience a small attraction from the optical trapof the other particle. After calibration, the depletion forceson the PS particles can be extracted from the particlepositions (see the Supplemental Material [64]). By tuningthe separation of two optical traps and the laser intensity,we obtain the FeffðrÞ as a function of r and the trapstiffness, respectively.Results and discussion.—We first perform simulations to

study the effect of constraint on FeffðrÞ. The dimensionlessstiffness coefficient, k ¼ kelσ2s=kBT, of the trap potential istuned from 0 (free) to infinity (frozen). In the simulations,we choose the packing fraction of active particles ρ ¼ 0.2to avoid the formation of large clusters, and the reducedself-propelling force Fdσs=kBT ¼ 20. Figure 2(a) plots themeasured Feff as a function of particle separation r andstiffness coefficient k. It is clear that the FeffðrÞ on thepassive particles strongly depends on the stiffness coef-ficient, in stark contrast to equilibrium cases wheredepletion forces are independent of constraints. For thefree passive particles, FeffðrÞ has a strong attraction at r ≃σl and a small peak of repulsion at r ≃ σl þ 0.8σs; while thefrozen case exhibits reverse properties, a weak attractionand a strong repulsive peak. This observation is consistentwith previous work, where free passive objects in an activebath aggregate together [44,54,56] while fixed ones exhibit

FIG. 1. (a) Sketch of simulation system, consisting of smallactive particles self-propelling toward the red side and two large(yellow) passive particles. Two vertical dashed lines mark threedifferent regions around the passive particles, in which the activeparticles contribute differently to depletion force. (b) Schematicexperimental system with PS particles (yellow) trapped opticallyin bacteria solution. The observing plane and trap centers aremarked by dashed lines.

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a strong repulsion [29,55,59,60]. The whole profile ofFeffðrÞ rises (becomes more repulsive) with k, and saturatesat the largest k [inset of Fig. 2(a)].The profile of FeffðrÞ and its dependence on constraints

can be understood by analyzing the distribution of activedepletants in different regions near the passive particles. Asillustrated in Fig. 1(a), collisions occurring in regions 1 and3 generate effective attractions between the passive spheresF1;3ðrÞ; while swimmers in region 2 tend to push the twopassive spheres apart, thus generating repulsions F2ðrÞ.The total depletion force is then the result of competition ofthe attractive and repulsive forces, FeffðrÞ¼F1;3ðrÞþF2ðrÞ.Moreover, due to their persistence of motion, the swimmersare more easily trapped and temporarily accumulated in theconcave region 2 than in the convex region 1 or 3[55,59,72]. At small r (≃σl), the concave area betweentwo passive particles is small with lower number ofswimmers in region 2 than region 1 or 3, and so theFeffðrÞ is attractive. The concave area increases with r,while the collision area in regions 1 and 3 remains constant.Thus, the repulsive force from the accumulation of activedepletants in region 2 eventually dominates the FeffðrÞ thatreaches a maximum at r < σl þ σs. Further increasing theseparation creates a gap that allows one swimmer to easilypass through without being trapped, which sharply reducesthe F2ðrÞ. For r ≤ σl þ 2σs, the gap between the passivespheres can be temporarily jammed by two swimmers witha certain probability, leading to an increase of F2ðrÞmanifested in the second and weaker peak in Fig. 2(a).At larger r, a dynamic bridge composed of multiple layersof swimmers cannot be maintained in the gap in dilutesuspensions [58]; the environments in the three regionsbecome equivalent, and the FeffðrÞ vanishes.

The constraints on the passive particles mainly influencethe repulsive force from region 2. Under weak or zeroconstraints, the swimmers in region 2 can easily pushthrough the gap between the passive objects, such that thecollision process is similar to that in regions 1 or 3. Understrong constraints, the passive spheres lack a sufficientdisplacement within the orientational diffusion time ofswimmer, which leads to a strong accumulation of theswimmers in the concave region 2. The accumulatedswimmers greatly enhance F2ðrÞ, and hence the totaleffective force becomes increasingly repulsive with k[Fig. 2(a)]. We verify the above scenario by directlyquantifying the repulsive (F2) and attractive (F1;3) forcesexerted on the passive particle and the correspondingswimmer packing fractions in region 2 and regions 1,3,as a function of the trap stiffness k, for passive particleseparation r=σs ¼ 3.1. As shown in Figs. 2(c) and 2(d), therepulsive force and the packing fraction in region 2 increasemore rapidly with k than their counterparts in regions 1,3,driving the overall active depletion force from attractive torepulsive. The transition stiffness coefficient, k ≃ 2000,where Feff changes the sign (F1;3 þ F2 ¼ 0), is higher thanthat inferred from the packing faction profiles, k ≃ 400,where ρ in regions 1,3 and region 2 become equal. Thisapparent discrepancy arises from the different collisionintensities on single swimmer level in regions 1,3 andregion 2. The swimmers colliding with the passive particlesin regions 1,3 are more likely to orientate parallel to theFeff , thus on average, generating a larger force per collision,which requires a higher swimmer concentration in region 2(larger k) to completely balance the F1;3.Our simulation results are verified by laser tweezer

experiments with PS particles and E. coli bacteria. In theexperiments, the dimensionless self-propelled force isestimated to be Fdσs=kBT ¼ 19.3 (σs the bacterial diam-eter), close to that in the simulations. The packing fractionof the bacteria near the focal plane is ρ ≃ 0.048.Considering the 3D experimental system, the evaluatedeffective 2D packing fraction is comparable to the simu-lation (see the Supplemental Material [64]). Figure 2(b)plots the measured effective interaction between the PSspheres as a function of particle separation and trapstiffness. The experiments demonstrate that the FeffðrÞincreases with the constraint, consistent with the simula-tions. Similar to the simulations, two repulsive peaks areobserved in the experiments for all three stiffness coef-ficients. The first peak corresponds to the maximum gap(r ≤ σl þ σs) between PS particles that forbids a bacteria topass. The peak values of the active depletion forces areseparately 2.1, 4.4, and 5.8 for k ≃ 67, 138, and 180,comparable to the simulation results for the same set of trapstiffness with peak values at 3.8, 5.1, and 5.4, respectively[inset of Fig. 2(a)]. The second peak corresponds to themaximum concave region (r ≤ σl þ 4σs) that does notallow the bacteria to pass through laterally (with its long

FIG. 2. Reduced effective force as a function of the distancebetween passive particles for various trap stiffness in simulation(a) and experiment (b), with negative forces being attraction. Theinset in (a) shows the maximum of Feff versus trap stiffness.(c) Magnitudes of F1;3 and F2, and (d) packing fractions ofswimmers in different regions versus trap stiffness.

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axis) but is freely permeable to the bacteria swimminglongitudinally, as the bacteria are rod shaped. The agree-ment between simulation results and experimental obser-vations suggests that hydrodynamic interactions, which islacking in the simulations, and the swimmer shape (spherein simulation, and rodlike in experiment) probably playlesser roles for active depletion interaction. To better mimicthe motion of bacteria in experiment, we perform simu-lations with the run-and-tumble dynamics, and obtainqualitatively the same results (see the SupplementalMaterial [64]).The strong constraint dependence of active depletion

force shows that the FeffðrÞ on free passive particles cannotbe obtained from frozen particles. We also examine thefeasibility of deriving the FeffðrÞ using kBT∇ ln½gðrÞ�, withgðrÞ the pair correlation function of the free passiveparticles in the active bath. The depletion force derivedfrom gðrÞ is plotted in Fig. 3(a) (pink line with circles) withthe thermal bath temperature, significantly different fromthat obtained from direct measurement (olive line). Thederived FeffðrÞ, however, exhibits a similar trend to themeasured force curve. It thus seems possible to employ aneffective temperature to scale the derived forces to the trueFeffðrÞ. The best-fitted profile yields an effective temper-ature around 0.6T and still shows clear discrepancies fromthe measured force near the attractive well (see theSupplemental Material [64]). An effective temperature lessthan T obviously disagrees with the conventional wisdomthat a passive particle in an active bath should have a highereffective temperature than in an equilibrium bath [31,32].Therefore, gðrÞ cannot be reliably employed to extract theactive depletion forces with a priori effective temperaturehigher than T, obtained independently from commonroutes. The pair correlation function can, however, quali-tatively determine the sign of active depletion forces forfree passive particles. The experimental gðrÞ of free PSparticles in an active bacterial bath shows a pronouncedpeak, indicating an attractive FeffðrÞ between the passivePS particles, in agreement with the simulations (see theSupplemental Material [64]).By exploring wider parameter spaces, we further dem-

onstrate that FeffðrÞ on free and fixed passive particles areessentially different (Fig. 3). Moreover, Figs. 3(a)–3(c) and3(e) show that the magnitudes of FeffðrÞ for both free andfixed cases increase with the swimmer driving force,packing fraction and rotational timescale (represented asthe Peclet number Pe ¼ ðFdγr=kBTσsγsÞwith constant Fd),as both F1;3ðrÞ and F2ðrÞ are, respectively, increasingfunctions of Fd, ρ and swimmer-particle interacting dura-tion. The increase of FeffðrÞ with depletant concentration isverified by experiments [Fig. 3(d)]. Although the γr and Fddependences of FeffðrÞ are similar, their effects on FeffðrÞare not quantitatively equivalent. For instance, the systemwith Fdσs=kBT ¼ 60 in Fig. 3(b) has Pe ¼ 20, but itsrepulsive peak is three times of its counterpart in Fig. 3(e).

This implies that the effects of γr and Fd cannot be reducedto a single Pe. The particle size dependence of FeffðrÞ isprovided in the Supplemental Material [64].We have shown that active depletion force on passive

particles strongly depends on their external elastic con-straints, which originates from the constraint-modifiedkinematics of passive spheres and (in turn) nearbyswimmers, facilitating the swimmer trapping in the regionbetween the passive particles under strong constraints.Besides the elastic trap, an alternative way to control thekinematics of colloidal body is to change its friction fromenvironment including boundaries and fluids. Similar to aparticle under a strong elastic trap, a passive particle with alarge friction coefficient cannot move fast enough inresponse to the impact of swimmers. Higher frictionalcoefficients of the particles thus effectively correspond tostronger constraints. In simulations, to exclusively study

FIG. 3. Reduced effective forces on free (a) and frozen(b) passive particles for different driving forces on swimmers,with ρ ¼ 0.2. The pink line with circles in (a) refers to FeffðrÞcalculated from gðrÞ with the thermal bath temperature T, and theright (red) vertical axis in (b) to the passive system. Panel (c) andits inset, respectively, denote FeffðrÞ on fixed and free passiveparticles for different swimmer packing fractions in simulations,with Fdσs=kBT ¼ 20. (d) Experimental FeffðrÞ for various ρ withk ¼ 137.8. Panel (e) and its inset separately denote FeffðrÞ onfixed and free passive particles for different γr (Pe), with Fd fixed.(f) FeffðrÞ on free passive particles of various frictional coef-ficients γl. The olive line is FeffðrÞ on fixed passive particles. Theinset shows the repulsive peak of FeffðrÞ versus γl, where thedashed line is a fit for extrapolation. In (e) and (f), Fdσs=kBT ¼20 and ρ ¼ 0.2.

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the effect of the kinematic constraint on FeffðrÞ, we changeγl independently while keeping γs constant [Fig. 3(f)]. Forcomparison, the FeffðrÞ for fixed case is also plotted, whichis effectively equivalent to infinite γl. With increasing γl, allthe features of FeffðrÞ under elastic constraints shown inFig. 2(a) are recovered, implying that the active depletionforces under both elastic and viscous constraints result fromsimilar underlying mechanisms. An interesting question ishow large the γl needs to be to observe FeffðrÞ on freeparticles with the same magnitude as that obtained in frozencases. A rough extrapolation of FeffðrÞ as a function of γlyields γl ¼ 6 × 104γs, which amounts to the frictionalcoefficient of a macroscopic sphere of radius 3 cm inwater. This further confirms that free mesoscopic colloidalparticles in active baths cannot be simply treated as fixedobjects.Conclusion.—By directly measuring the forces in steady

states, we show that the active depletion forces on passivecolloidal objects in an active bath are greatly influenced bythe elastic or kinematic constraints on the objects. Theeffective interaction changes from a strong attraction to astrong repulsion as the constraint increases. Frozen par-ticles cannot be employed as force probes to estimate theactive depletion forces on freely moving passive particles.The pair correlation function, which is directly related toeffective interparticle potential in equilibrium systems, alsofails to quantitatively extract the active depletion forces.The key features of the simulation results are verified inexperiments with PS particles optically trapped in bacteriasolutions, wherever comparison is possible. Therefore, theactive depletion interactions must be properly taken intoaccount, when designing or analyzing activity-directedassembly, phase transition, and transportation of colloids.Our work can be straightforwardly extended to studyeffective interactions in other diverse active systems.

We thank K. Zhao and Y.W. Zong at Tianjin Universityfor providing the E. coli bacteria. We acknowledge supportfrom National Natural Science Foundation of China(Grants No. 11874397, No. 11874395, No. 11674365,No. 11474327, No. 11475018, and No. 11774394). Thiswork was also supported by the MOST 973 Program(No. 2015CB856800) and the Key Research Program ofFrontier Sciences of Chinese Academy of Sciences (GrantNo. QYZDB-SSW-SYS003).

*These authors contributed equally to this work.†fye@iphy.ac.cn‡kechen@iphy.ac.cn§mcyang@iphy.ac.cn

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